When we further take \(C_{0,0} :=\alpha _0 I\) , \(C_{1,0} := \beta _0 C\)  for \(\alpha _0, \beta _0 \in \mathbb {C}\) , \(C \in {\mathcal {L}}(Y_1, X)\) , and \(c_{0,1}:=\alpha _1 I\)  and \(c_{1,1}:= \beta _1 C\)  for \(\alpha _1, \beta _1 \in S^{\prime }\) , then the convolution term \(c\ast u\)  corresponds to the one from the heat-conduction problem with memory [1]}.
Such form of the convolutor \(c\)  was considered in the context of (REF ) in [2]}. In this scalar case one can easily specify the assumptions on the functions \(\hat{\alpha }_1\) , \(\hat{\beta }_1\)  and the operator \(C\) , which imply that \(c\)  satisfies the assumptions of Theorem REF ; cf. [2]} and references therein.
